3.1995 \(\int \frac{(2+3 x)^5 (3+5 x)^2}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=105 \[ \frac{405}{128} (1-2 x)^{15/2}-\frac{97605 (1-2 x)^{13/2}}{1664}+\frac{672003 (1-2 x)^{11/2}}{1408}-\frac{285565}{128} (1-2 x)^{9/2}+\frac{842415}{128} (1-2 x)^{7/2}-\frac{1623419}{128} (1-2 x)^{5/2}+\frac{6206585}{384} (1-2 x)^{3/2}-\frac{2033647}{128} \sqrt{1-2 x} \]

[Out]

(-2033647*Sqrt[1 - 2*x])/128 + (6206585*(1 - 2*x)^(3/2))/384 - (1623419*(1 - 2*x
)^(5/2))/128 + (842415*(1 - 2*x)^(7/2))/128 - (285565*(1 - 2*x)^(9/2))/128 + (67
2003*(1 - 2*x)^(11/2))/1408 - (97605*(1 - 2*x)^(13/2))/1664 + (405*(1 - 2*x)^(15
/2))/128

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Rubi [A]  time = 0.0801906, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{405}{128} (1-2 x)^{15/2}-\frac{97605 (1-2 x)^{13/2}}{1664}+\frac{672003 (1-2 x)^{11/2}}{1408}-\frac{285565}{128} (1-2 x)^{9/2}+\frac{842415}{128} (1-2 x)^{7/2}-\frac{1623419}{128} (1-2 x)^{5/2}+\frac{6206585}{384} (1-2 x)^{3/2}-\frac{2033647}{128} \sqrt{1-2 x} \]

Antiderivative was successfully verified.

[In]  Int[((2 + 3*x)^5*(3 + 5*x)^2)/Sqrt[1 - 2*x],x]

[Out]

(-2033647*Sqrt[1 - 2*x])/128 + (6206585*(1 - 2*x)^(3/2))/384 - (1623419*(1 - 2*x
)^(5/2))/128 + (842415*(1 - 2*x)^(7/2))/128 - (285565*(1 - 2*x)^(9/2))/128 + (67
2003*(1 - 2*x)^(11/2))/1408 - (97605*(1 - 2*x)^(13/2))/1664 + (405*(1 - 2*x)^(15
/2))/128

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Rubi in Sympy [A]  time = 11.2505, size = 94, normalized size = 0.9 \[ \frac{405 \left (- 2 x + 1\right )^{\frac{15}{2}}}{128} - \frac{97605 \left (- 2 x + 1\right )^{\frac{13}{2}}}{1664} + \frac{672003 \left (- 2 x + 1\right )^{\frac{11}{2}}}{1408} - \frac{285565 \left (- 2 x + 1\right )^{\frac{9}{2}}}{128} + \frac{842415 \left (- 2 x + 1\right )^{\frac{7}{2}}}{128} - \frac{1623419 \left (- 2 x + 1\right )^{\frac{5}{2}}}{128} + \frac{6206585 \left (- 2 x + 1\right )^{\frac{3}{2}}}{384} - \frac{2033647 \sqrt{- 2 x + 1}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**5*(3+5*x)**2/(1-2*x)**(1/2),x)

[Out]

405*(-2*x + 1)**(15/2)/128 - 97605*(-2*x + 1)**(13/2)/1664 + 672003*(-2*x + 1)**
(11/2)/1408 - 285565*(-2*x + 1)**(9/2)/128 + 842415*(-2*x + 1)**(7/2)/128 - 1623
419*(-2*x + 1)**(5/2)/128 + 6206585*(-2*x + 1)**(3/2)/384 - 2033647*sqrt(-2*x +
1)/128

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Mathematica [A]  time = 0.0572894, size = 48, normalized size = 0.46 \[ -\frac{1}{429} \sqrt{1-2 x} \left (173745 x^7+1002375 x^6+2632743 x^5+4212525 x^4+4694340 x^3+4058988 x^2+3152152 x+3275704\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((2 + 3*x)^5*(3 + 5*x)^2)/Sqrt[1 - 2*x],x]

[Out]

-(Sqrt[1 - 2*x]*(3275704 + 3152152*x + 4058988*x^2 + 4694340*x^3 + 4212525*x^4 +
2632743*x^5 + 1002375*x^6 + 173745*x^7))/429

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Maple [A]  time = 0.007, size = 45, normalized size = 0.4 \[ -{\frac{173745\,{x}^{7}+1002375\,{x}^{6}+2632743\,{x}^{5}+4212525\,{x}^{4}+4694340\,{x}^{3}+4058988\,{x}^{2}+3152152\,x+3275704}{429}\sqrt{1-2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+3*x)^5*(3+5*x)^2/(1-2*x)^(1/2),x)

[Out]

-1/429*(173745*x^7+1002375*x^6+2632743*x^5+4212525*x^4+4694340*x^3+4058988*x^2+3
152152*x+3275704)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.3435, size = 99, normalized size = 0.94 \[ \frac{405}{128} \,{\left (-2 \, x + 1\right )}^{\frac{15}{2}} - \frac{97605}{1664} \,{\left (-2 \, x + 1\right )}^{\frac{13}{2}} + \frac{672003}{1408} \,{\left (-2 \, x + 1\right )}^{\frac{11}{2}} - \frac{285565}{128} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{842415}{128} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{1623419}{128} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{6206585}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{2033647}{128} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(3*x + 2)^5/sqrt(-2*x + 1),x, algorithm="maxima")

[Out]

405/128*(-2*x + 1)^(15/2) - 97605/1664*(-2*x + 1)^(13/2) + 672003/1408*(-2*x + 1
)^(11/2) - 285565/128*(-2*x + 1)^(9/2) + 842415/128*(-2*x + 1)^(7/2) - 1623419/1
28*(-2*x + 1)^(5/2) + 6206585/384*(-2*x + 1)^(3/2) - 2033647/128*sqrt(-2*x + 1)

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Fricas [A]  time = 0.213941, size = 59, normalized size = 0.56 \[ -\frac{1}{429} \,{\left (173745 \, x^{7} + 1002375 \, x^{6} + 2632743 \, x^{5} + 4212525 \, x^{4} + 4694340 \, x^{3} + 4058988 \, x^{2} + 3152152 \, x + 3275704\right )} \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(3*x + 2)^5/sqrt(-2*x + 1),x, algorithm="fricas")

[Out]

-1/429*(173745*x^7 + 1002375*x^6 + 2632743*x^5 + 4212525*x^4 + 4694340*x^3 + 405
8988*x^2 + 3152152*x + 3275704)*sqrt(-2*x + 1)

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Sympy [A]  time = 24.5756, size = 94, normalized size = 0.9 \[ \frac{405 \left (- 2 x + 1\right )^{\frac{15}{2}}}{128} - \frac{97605 \left (- 2 x + 1\right )^{\frac{13}{2}}}{1664} + \frac{672003 \left (- 2 x + 1\right )^{\frac{11}{2}}}{1408} - \frac{285565 \left (- 2 x + 1\right )^{\frac{9}{2}}}{128} + \frac{842415 \left (- 2 x + 1\right )^{\frac{7}{2}}}{128} - \frac{1623419 \left (- 2 x + 1\right )^{\frac{5}{2}}}{128} + \frac{6206585 \left (- 2 x + 1\right )^{\frac{3}{2}}}{384} - \frac{2033647 \sqrt{- 2 x + 1}}{128} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+3*x)**5*(3+5*x)**2/(1-2*x)**(1/2),x)

[Out]

405*(-2*x + 1)**(15/2)/128 - 97605*(-2*x + 1)**(13/2)/1664 + 672003*(-2*x + 1)**
(11/2)/1408 - 285565*(-2*x + 1)**(9/2)/128 + 842415*(-2*x + 1)**(7/2)/128 - 1623
419*(-2*x + 1)**(5/2)/128 + 6206585*(-2*x + 1)**(3/2)/384 - 2033647*sqrt(-2*x +
1)/128

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GIAC/XCAS [A]  time = 0.210944, size = 155, normalized size = 1.48 \[ -\frac{405}{128} \,{\left (2 \, x - 1\right )}^{7} \sqrt{-2 \, x + 1} - \frac{97605}{1664} \,{\left (2 \, x - 1\right )}^{6} \sqrt{-2 \, x + 1} - \frac{672003}{1408} \,{\left (2 \, x - 1\right )}^{5} \sqrt{-2 \, x + 1} - \frac{285565}{128} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{842415}{128} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{1623419}{128} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{6206585}{384} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{2033647}{128} \, \sqrt{-2 \, x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x + 3)^2*(3*x + 2)^5/sqrt(-2*x + 1),x, algorithm="giac")

[Out]

-405/128*(2*x - 1)^7*sqrt(-2*x + 1) - 97605/1664*(2*x - 1)^6*sqrt(-2*x + 1) - 67
2003/1408*(2*x - 1)^5*sqrt(-2*x + 1) - 285565/128*(2*x - 1)^4*sqrt(-2*x + 1) - 8
42415/128*(2*x - 1)^3*sqrt(-2*x + 1) - 1623419/128*(2*x - 1)^2*sqrt(-2*x + 1) +
6206585/384*(-2*x + 1)^(3/2) - 2033647/128*sqrt(-2*x + 1)